Can you please help me with this problem:
The x-intercept of a line has a value of -7/6 and a y-intercept of 7. Another line has a slope that is twice the slope of the first line an a y-intercept that is 15 more than the first lines' y-intercept. What is the product of the coordinates in the solution to this system? Write a system of equations for this situation and find its solution.
This is what I have:
x = -7/6 which is -1.16 - twice this amount is 2.33
y = 7 (15 more than that first y intercept is 22)
coordinates are 2.33 and 22...is this correct? Thank you
First of all , don't switch to decimals
( 7/6 ≠ 1.16, it is 1.1666.... )
So you have two points (-7/6 , 0) and (0,7)
slope = (7-0)/(0+7/6)
= 7/(7/6) = 6
equation of first line : y = 6x + 7
slope of 2nd line = 12 <----- it said twice the first
y-intercept of 2nd line = 7+15 = 23
so for the second line
equation is y = 12x + 23
solve them:
12x+23 = 6x+7
6x = -16
x = -16/6 = -8/3
then y = 6(-8/3) + 7 = -9
so the product of the coordinates = (-8/3)(-9)
= 24
To find the solution to this system of equations, let's first write down the equations for the given situation.
Let's assume the equation of the first line is y = mx + b, where m is the slope and b is the y-intercept.
From the given information, we know that the x-intercept of the first line is -7/6 and the y-intercept is 7.
The x-intercept occurs when y = 0. So, substituting these values in the equation, we get:
0 = m(-7/6) + b ----(1)
Similarly, the second line has a slope twice that of the first line and a y-intercept that is 15 more than the first line's y-intercept. Let's denote the slope of the second line as 2m (twice the slope of the first line). The y-intercept of the second line will be b + 15.
So, the equation of the second line becomes:
y = 2m(x) + (b + 15) ----(2)
Now, we have a system of equations. Let's substitute the values we know into these equations:
From equation (1), we know that the x-intercept is -7/6, so we have:
0 = m(-7/6) + b
Simplifying this equation, we get:
0 = -7m/6 + b ----(3)
From the problem, we also know that the y-intercept of the first line is 7, so we have:
b = 7
Substituting this value in equation (3), we get:
0 = -7m/6 + 7 ----(4)
Now, from equation (2), we know that the slope of the second line is twice that of the first line, which means 2m. And the y-intercept is (b+15), so we have:
y = 2m(x) + (b + 15)
Substituting the known values, we get:
y = 2m(x) + (7 + 15)
Simplifying this equation, we get:
y = 2m(x) + 22 ----(5)
We now have a system of equations:
Equation (4): 0 = -7m/6 + 7
Equation (5): y = 2m(x) + 22
To find the solution to the system, we need to solve for the values of m and x.
From equation (4), we can solve for m:
-7m/6 + 7 = 0
Multiplying both sides of the equation by 6, we get:
-7m + 42 = 0
Simplifying, we find:
-7m = -42
Dividing both sides by -7, we get:
m = 6
Now, substitute the value of m into equation (5):
y = 2(6)(x) + 22
Simplifying, we get:
y = 12x + 22
We have found the equation of the second line.
To find the coordinates of the solution, we need to find the intersection point of the two lines. This occurs when the y-values are equal.
So, we set the equations equal to each other and solve for x:
12x + 22 = 0
Subtracting 22 from both sides, we get:
12x = -22
Dividing both sides by 12, we find:
x = -22/12 = -11/6
Therefore, the x-coordinate of the intersection point is -11/6.
To find the y-coordinate, substitute this value back into either equation (4) or (5). Let's use equation (5):
y = 12(-11/6) + 22
Simplifying, we get:
y = -22 + 22
So, the y-coordinate of the intersection point is 0.
The coordinates of the solution are (-11/6, 0).
Now, to find the product of the coordinates, multiply the x-coordinate and the y-coordinate:
(-11/6) * 0 = 0
Therefore, the product of the coordinates in the solution to this system is 0.
I hope this helps! Let me know if you have any further questions.